Skip to main content
SpringerLink
Log in

Wild singularities of flat surfaces

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We consider flat surfaces and the points of their metric completions, particularly the singularities to which the flat structure of the surface does not extend. The local behavior near a singular point x can be partially described by a topological space L(x) which captures all the ways that x can be “approached linearly”. The homeomorphism type of L(x) is an affine invariant. When x is not a cone point or an infinite-angle singularity, we say it is wild; in this case it is necessary to add further metric data to L(x) to get a quantitative description of the surface near x.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. J. P. Bowman, The complete family of Arnoux-Yoccoz surfaces, Geometriae Dedicata, posted on 2012, 1–18, DOI 10.1007/510711-012-9762-9, to appear.

  2. J. P. Bowman, Finiteness conditions on translation surfaces, Quasiconformal Mappings, Riemann Surfaces, and Teichmüller Spaces, 2012, pp.

  3. D. Burago, Yu. Burago and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, Vol. 33, American Mathematical Society, Providence, RI, 2001.

    MATH  Google Scholar 

  4. R. Chamanara, Affine automorphism groups of surfaces of infinite type, in In the Tradition of Ahlfors and Bers, III, Contemporary Mathematics, Vol. 355, American Mathematical Society, Providence, RI, 2004, pp. 123–145.

    Chapter  Google Scholar 

  5. R. Chamanara, F. Gardiner and N. Lakic, A hyperelliptic realization of the horseshoe and baker maps, Ergodic Theory and Dynamical Systems 26 (2006), 1749–1768.

    Article  MathSciNet  MATH  Google Scholar 

  6. A. de Carvalho and T. Hall, Paper folding, Riemann surfaces, and convergence of pseudo-Anosov sequences, Geometry & Topology 16 (2012), 1881–1966.

    Article  MathSciNet  MATH  Google Scholar 

  7. R. H. Fox and R. B. Kershner, Concerning the transitive properties of geodesics on a rational polyhedron, Duke Mathematical Journal 2 (1936), 147–150.

    Article  MathSciNet  Google Scholar 

  8. E. Ghys, Topologie des feuilles génériques, Annals of Mathematics 141 (1995), 387–422.

    Article  MathSciNet  MATH  Google Scholar 

  9. E. Gutkin and C. Judge, Affine mappings of translation surfaces: geometry and arithmetic, Duke Mathematical Journal 103 (2000), 191–213.

    Article  MathSciNet  MATH  Google Scholar 

  10. P. Hooper, The invariant measures of some infinite interval exchange maps, (2010), available at arXiv:1005.1902v1.

  11. P. Hubert S. Lelièvre and S. Troubetzkoy, The Ehrenfest wind-tree model: periodic directions, recurrence, diffusion, Journal für die Reine und Angewandte Mathematik (Crelle’s Journal) 656 (2011), 223–224.

    MATH  Google Scholar 

  12. P. Hubert and G. Weize-Schmithüsen, Infinite translation surfaces with infinitely generated Veech groups, Journal of Modern Dynamics 4 (2010), 715–732

    Article  MathSciNet  MATH  Google Scholar 

  13. P. Przytycki, G. Schmithüsen and F. Valdez, Veech groups of Loch Ness Monsters, Annales de l’Institut Fourier 61 (2011), 673–687.

    Article  MATH  Google Scholar 

  14. F. Valdez, Veech groups, irrational billiards and stable abelian differentials, Discrete and Continuous Dynamical Systems, Series A (2012), 1055–1063.

Download references

Author information

Authors and Affiliations

  1. IMS, Stony Brook University, Stony Brook, NY, 11794-3660, USA

    Joshua P. Bowman

  2. Centro de Ciencias Matemáticas, U.N.A.M., Campus Morelia, Michoacán, México

    Ferrán Valdez

Authors
  1. Joshua P. Bowman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ferrán Valdez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joshua P. Bowman.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bowman, J.P., Valdez, F. Wild singularities of flat surfaces. Isr. J. Math. 197, 69–97 (2013). https://doi.org/10.1007/s11856-013-0022-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-013-0022-y

Keywords

Search

Navigation